GEORGIAN MAHEMAICAL JOURNAL: Vol. 6, No. 1, 1999, 91-98 WEAKLY PERIODIC SEQUENCES OF BOUNDED LINEAR RANSFORMAIONS: A SPECRAL CHARACERIZAION A. R. SOLANI AND Z. SHISHEBOR Abstract. Let X and Y be two Hilbert spaces, and LX, Y ) the space of bounded linear transformations from X into Y. Let {A n } LX, Y ) be a weakly periodic sequence of period. Spectral theory of weakly periodic sequences in a Hilbert space is studied by H. L. Hurd and V. Mandrekar 1991). In this work we proceed further to characterize {A n } by a positive measure µ and a number of LX, X)-valued functions a,..., a ; in the spectral form A n 2π e iλn 2πkn Φdλ)V n λ), where V nλ) e i a k k λ) and Φ is an LX, Y )-valued Borel set function on [, 2π) such that Φ )x, Φ )x ) Y x, x ) X µ ). 1. Introduction Let X and Y be Hilbert spaces, and let LX, Y ) stand for bounded linear transformations from X into Y. Suppose {A n, n Z}, Z is the set of integers, is a sequence in LX, Y ). he sequence {A n } is said to be weakly periodic, if for every n, m Z, A n x, A m x ) Y A n x, A m x ) Y, x, x X, for some >. he smallest indicates the period, and in this case we use the notation -WP. We call the sequence {A n } LX, Y ) strongly periodic if A n A n, n Z. Such a sequence is denoted by -SP. Every 1-WP is stationary. Spectral theory of -WP sequences in Hilbert spaces, and in certain Banach spaces, was first studied by H. L. Hurd and V. Mandrekar [1]. If X is finite dimensional and Y L 2 Ω, F, P ), where Ω, F, P ) is a probability space, then every -WP sequence is a periodically correlated second order process. Such a process was first introduced and studied by Gladyshev [2]. For more on these kind of processes see [3], [4]. 1991 Mathematics Subject Classification. Primary 47B15, 47B99; Secondary 6G12. Key words and phrases. Hilbert space, bounded linear operators, weakly periodic sequences, spectral representation. 91 172-947X/99/1-91$16./ c 1999 Plenum Publishing Corporation
92 A. R. SOLANI AND Z. SHISHEBOR he significance of a spectral representation for a given sequence is realized in probability theory. It amounts to considering the elements of the sequence as the Fourier coefficients of a certain random) measure. Let Ψ be a set function on Borel sets in [, 2π) with values in LX, Y ). Among the following properties; i) for every x X, Ψ )x is an Y -valued measure, ii) for every x, x X, Ψ )x, Ψ )x ) Y if and Ψ )x, Ψ )x ) Y defines a complex measure on [, 2π); if i) is satisfied, then we call Ψ a random spectrum, RS; if i) and ii) both are satisfied, then we call Ψ a random spectrum with orthogonal increments, RSOI. If Ψ is a RS, then B n is well-defined in the sense that B n x, B m x ) Y e inλ Ψdλ), n Z, 1.1) e inλ mλ ) Ψdλ)x, Ψdλ )x ) Y e inλ mλ ) Γ x,x dλ, dλ ), where Γ x,x, ) is a bimeasure on [, 2π) [, 2π). Any sequence {B n } LX, Y ) given by 1.1) is called harmonizable [4]. Every harmonizable sequence in LX, Y ), for which its Ψ also satisfies ii), is a stationary sequence [5]. By using a certain th root of a unitary operator Hurd and Mandrekar [1] provided the Gladyshev s type representation for a -WP sequence in a Hilbert space. It can be deduced from their work that {A n } LX, Y ) is a -WP sequence if and only if A n e inλ Zdλ), n Z, where Z is a RS on [, 2π), that its restriction to each [ ), p,..., 1, is a RSOI; more precisely, Z ) W p [ p W p, 2πp1) )x, W q 2πq x )) Y 2πp 1) )),, 1.2) for every, [, 2π ) with, and every p, q,..., 1. his representation was also independently derived by the authors in [6] by using a different approach. In this work we proceed further to identify each -WP sequence {A n } in LX, Y ) with a positive measure µ as a number of LX, X)-valued functions, a,..., a. he identification is given in heorem 3.4, which
A SPECRAL CHARACERIZAION 93 is a refinement of Gladyshev s result on periodically correlated sequences. he relationship between the functions a,..., a and the matrix f, given in 2.3), introduced by Gladyshev is also provided in heorem 3.4. An application of our representation to the regularity of periodically correlated sequences is as follows. A periodically correlated sequence X n, n Z, is regular, i.e., span{x t, n t n} {}, if and only if / log det fλ) dλ >, with f as in 2.3) see [2]). It follows from 3.3) that det fλ) ) 2, j a λ 2πj λ [, 2π/ ). herefore the sequence is regular if and only if log a λ) dλ >. heorem 3.1 in [3] easily follows from this observation, as f in [3] dominates a. We summarize our characterization by the spectral representation A n 2π e inλ Φdλ)V n λ), where, Φ is a certain RSOI, as in the abstract, and for each λ [, 2π), {V n λ)} LX, X) is a -SP in n, as is defined in heorem 3.4. 2. Notation and Preliminaries Suppose Z is given by 1.4); let F be the LX, X)-valued Borel set function on [, 2π) [, 2π) for which x, F, )x ) X Z )x, Z )x ) Y, x, x X. 2.1) We call F the spectral measure of the sequence {A n }. Note that the support of F lies on the lines d k : s t 2πk, k 1,..., 1. Furthermore A n x, A m x ) Y e intims x, F dt, ds)x ) X, x, x X. Let F k be the restriction of F to the line d k, k 1,...,. Let µ be a positive finite measure for which each F k is µ-continuous and µ 2πk dx) µdx) on [, 2π/ ), k 1,..., 1, 2.2) and x, F k )x ) x, f ks)x )µds), where f k are LX, X)-valued functions on [, 2π), k 1,..., 1, [7]. o see the existence of such a measure, note that corresponding to each F k there is a positive finite measure µ k such that F k is µ k -continuous [8]. Now let ν k1 µ k, and ν 2πk µe) j1 k1 ν 2πk Now fs) [f p l s ) ] E [ 2πj) E) νe 2πk, 2πj )). [, s l,p,..., ) [, 2π)), then define, 2π ), 2.3) defines an operator-matrix; we refer to a matrix whose entries are operators as an operator-matrix. We call f the spectral density of {A n } with respect
94 A. R. SOLANI AND Z. SHISHEBOR to µ. Recall that an operator-matrix [ i,j ] i, j,..., 1 is called positive definite if and only if the matrix [x kj, j,l x kl )] j,l,...,, is positive definite for each x k,..., x k. We recall that L 2 µ, X) stands for µ- Bochner integrable functions g :, 2π] X such that gλ) 2 X µdλ) < the proof of which is omitted [9]. 3. Spectral Characterization H n integral stochastic integral) of the form Φdλ)gλ), where g L 2 µ, X) and Φ is the RSOI given below in 3.1), can be defined by using the well known result on a existence of the weak stochastic integral. he detail is given in the following theorem. For more on a stochastic integral we refer the readers to [1] and [1]. heorem 3.1. Let µ be a positive finite Borel measure on [, 2π) and Φ be a RSOI for which Φ )x, Φ )x ) Y x, x ) X µ ). 3.1) hen for every g L 2 µ, X), Φdλ)gλ) is well defined as an element in Y and 2π { } Φdλ)gλ) Y gλ) 2 2 1 Xµdλ). Moreover, the integral satisfies the following properties: i) Φdλ)g 1 λ)g 2 λ)) Φdλ)g 1 λ) Φdλ)g 2 λ), g 1, g 2 L 2 µ, X); ii) if g n g in L 2 µ, X), then Φdλ)g n λ) Φdλ)gλ) in Y as n. Corollary 3.2. Let µ and Φ be as in heorem 3.1. Let { } A a; a : [, 2π) LX, X), aλ)x Xµdλ) 2 <, x X. hen for every a A, x X, Φdλ)aλ)x is well defined as an element in Y, and 2π { 2π } 1 Φdλ)aλ)x aλ)x 2 2 Xµdλ). Y Moreover, for every x, x X and every a, b A, i) Φdλ)aλ) bλ))x Φdλ)aλ)x Φdλ)bλ) x, ii) Φdλ)aλ)x x ) Φdλ)aλ)x Φdλ)aλ)x, iii) for a given x X, if a n )x a )x in L 2 µ, X), then Φdλ)a n λ)x Φdλ)aλ)x in Y.
A SPECRAL CHARACERIZAION 95 Proof. For each fix x X, a )x is an X-valued function. If a A, then a )x L 2 µ, X). Now apply heorem 3.1 and use the linear property of a ). he following lemma concerns the Cholesky decomposition for a positive definite operator-matrix Lemma 3.5 in [11]) and it plays a crucial role in our characterization. Lemma 3.3. Let M n be an n n positive operator-matrix, then M n U n U n, where U n is an n n upper triangular matrix. he following theorem is the essential result of this paper. heorem 3.4. Let {A n } LX, Y ). hen {A n } is a -WP if and only if A n e ins Φds)V n s), 3.2) where i) Φ is a RSOI that satisfies 3.1) and µ is a finite measure satisfying 2.2); ii) V n s) 2πkn k e i a k s 2πk ), s [, 2π), n Z, where each a k is an LX, X)-valued function on [, 2π) with a k s) for s [, 2πk ). Furthermore, the triangular operator-matrix Ax) [a 2πj j k x )] k j, k, j,..., 1, satisfies fx) A x)ax), [ x, 2π ), 3.3) i.e., it is the Cholesky decomposition of the density fx) given by 2.3), a x) stands for the adjoint of ax). Proof. Let {A n } be a sequence for which 3.2) holds, i.e., hus k l A n k e ins Φ 2πk ds)a k s). 3.4) A n x, A m x ) Y e insimt Φ 2πk ds)a k s)x, Φ 2πl dt)a l t)x ) Y.3.5)
96 A. R. SOLANI AND Z. SHISHEBOR It also follows from 3.1) that Φ 2πk ds)a k s)x, Φ 2πl dt)a l t)x ) Y { ak s)x, a pk s )x ) X µ ds), s 2πk t 2πl, otherwise, where p l k. Since µ ds) µds), p 1,..., 1. We obtain p ) p1 e e insims ) A n x, A m x ) Y k p e insims k p insims ) ak s)x, a kp s/ )x ) X µds) a k s)x, a k s)x ) X µds) ak s)x, a kp s / )x ) µds). 3.6) X k It is clear from 3.6) that {A n } is a -WP sequence. Conversely, let {A n } be a -WP sequence; then it follows from 2.1) that A n x, A m x ) Y p l e insimt Zds)x, Zdt)x ) Y e insimt x, F ds, dt)x ) X p1) p1) k 1 p p1) l1) 2πl e insimt x, F ds, dt)x ) X 2πp l)m in m)s i e x, F p l ds)y) X p l k p1) 2πkm in m)s i p l k1 pk p1) e e in m)s x, f ds)y) X µds) x, f k ds)y) X µds) 2πkm im n)s i e x, f k ds)y) X µds)
A SPECRAL CHARACERIZAION 97 k 1 k1 k) 2πmk i e e in m)s x, f k ds)y) X µds) e in m)s x, f ds)y) X µds) 2πmk i e e in m)s x, f k ds)y) X µds). 2πk By using 2.3) and comparing the last equality with 3.6) we obtain that if a k, k 1,..., 1, satisfy k p a k s)x, a pk s ) x ) X x, f p s)x ) X, p 1,...,, a k s)x, a k s)x ) X x, f s)x ) X, 3.7) k p k a k s)x, a kp s ) x ) X x, f p s)x ) X, p 1,..., 1, then {A n } will satisfy 3.2), as 3.2), 3.4), 3.5) and 3.6) are equivalent. But it easily follows that 3.7) and 3.3) are identical. On the other hand, 3.3) follows by applying Lemma 3.3 to f. Remark 3.5. We were led by the referee to the representation that {A n } LX, Y ) is -WP if and only if A n e iλn Ψdλ)D n, 3.8) where Ψ is an LX, Y )-valued measure and {D n } LX, X) is a -SP sequence. In 3.8) Ψ lacks property 3.1), but instead each {D n } is independent of λ. Let us provide the referee s proof which is a nice application of the symbolic calculus to unitary operators. Proof. Since {A n } is -weakly periodic, A n UA n for some unitary U in Y and each n and. We define B n U n/ A n. It is easy to make sure that {B n } LX, Y ) is -strongly periodic and A n U n/ B n. Writing the spectral representation for the unitary operator U / we get A n e iλn Edλ)B n, where Edλ) is a solution of the identity of Y. Now if X is infinite-dimensional, one can establish an isomorphism without loss of generality one can assume that X and Y are separable) j LY, X) and we come to 3.8) with Ψdλ) Edλ)j and D n jb n.
98 A. R. SOLANI AND Z. SHISHEBOR Acknowledgement his research was supported by the AEOI, Center for heoretical Physics and Mathematics. he authors would like to thank the referee for reading carefuuly the paper and providing valuable comments such as the representation cited in Remark 3.5. References 1. H. L. Hurd and V. Mandrekar, Spectral theory of periodically and quasi-periodically stationary SαS sequences. ech. Report No. 349, Center for Stochastic Processes, Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-326, USA, 1991. 2. E. G. Gladyshev, Periodically correlated random sequences. Soviet Math. Dokl. 21961), 385 388. 3. A. G. Miamee, Periodically correlated processes and their stationary dilations. SIAM J. Appl. Math. 5199), 1194 1199. 4. M. Pourahmadi and H. Salehi, On subordination and linear transformation of harmonizable and periodically correlated processes. Probability heory on Vector Spaces, III Lublin), Lecture Notes in Math., 18, 195 213, Springer, Berlin New York, 1983. 5. Yu. A. Rozanov, Some approximation problems in the theory of stationary processes. J. Multivariate Anal. 21972), 135 144 6. Z. Shishebor and A. R. Soltani, A spectral representation weakly periodic sequences of bounded linear transformations. echnical report, Department of Statistics, Shiraz University, Shiraz, Iran, 1995. 7. V. Mandrekar and H. Salehi, On singularity and Lebesgue type decomposition for operator-valued measures. J. Multivariate Anal. 21971), 167 185. 8. N. Dunford and J.. Schwartz, Linear operators, I. Interscience, New York, 1958. 9. J. Diestel and J. J. Uhl, Vector measures. Amer. Math. Soc., Providence, R.I., 1977. 1. Yu. A. Rozanov, Stationary random processes. Holden-Day, San Francisco, 1967. 11. G.. Adams, P. J. McGuire, and V. I. Paulsen, Analytic reproducing kernels and multiplication operators. Illinois J. Math. 361992), 44 419. Received 18.12.1996; revised 5.9.1997) Center for heoretical Physics and Mathematics AEOI P.O. Box 11365-8486, ehran Iran Authors addresses: Department of Statistics Shiraz University Shiraz 71454 Iran